From Within
Fallout 4 Guide

How to unlock: Talk to elder Maxson in Boston Airport (M7,3) or on the Prydwen ship. The quest becomes available once you build the teleport during The Molecular Level quest.

Quest objectives:

Locate Doctor Li

Speak to Doctor Li in Advanced Systems

Retrieve Evidence to convince Doctor Li

Bring the Holotape to Doctor Li

Report to Elder Maxson

Find doctor Li

You receive the From Within quest from elder Maxson - the leader of the Brotherhood of Steel in the Commonwealth. However, it will be possible only after you build the teleport in The Molecular Level quest of Brotherhood's main storyline.

Your task is to find doctor Li who escaped from the Brotherhood of Steel some time ago and joined the Institute. You must convince her to return. Once you reach the Institute, keep going ahead and drive down the elevator. Go to the main square of the complex after leaving Shaun'sroom. Now find Advanced Systems building and enter it. Doctor Li can be found in one of the rooms.

You can use your charisma to convince her to return. Sadly, even if you have the statistic developed to maximum, there is a large chance that you will fail in doing it. Luckily, there is another possibility. Doctor Li wants to know what happened with her colleague. The level on which he was working has been shut down. You must get there and investigate.

Obtain the evidence needed for convincing doctor Li

Hacking the master terminal will allow you to complete the mission much faster.

Leave the building and go to the Biosciencebuilding from the main square of the institute. To get to the locked level you can walk through the passage blocked with lasers. To do it, hack the nearby terminal (master level). This will allow you to avoid walking through the whole locked laboratory.

The second way is breaking into the room in the eastern side of the building (novice level lock). Once you're inside, you must keep walking ahead. Watch out for the automatic turrets and traps. Few times a terminal or a lock can stand on your way (novice level). Finally you will reach the room with capsules containing supermutants. Next to the computer you will find the holotape which you must take.

Now go back to doctor Li in Advanced Systems. After hearing the tape she will agree on returning to the Brotherhood. Now go to elder Maxson on the Prydwen ship. For completing the mission you will receive some experience.